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Equilibrium Asset Pricing with Transaction Costs

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Summary

This paper represents the major theoretical advance over the earlier Herdegen-Muhle-Karbe (2018) paper by making the equilibrium volatility endogenous. Two agents with mean-variance preferences and heterogeneous risk aversions trade a risky asset subject to quadratic transaction costs. The equilibrium price, expected return, AND volatility are all determined simultaneously via market clearing.

The key technical challenge is that with endogenous volatility, the FBSDEs characterising the equilibrium (equations 4.6-4.8) become fully coupled and nonlinear — the backward equation for the trading rate depends quadratically on the forward volatility process, while the volatility itself is determined by a BSDE coupled to the trading rate. This is fundamentally harder than the exogenous-volatility case where FBSDEs are linear.

The main existence result (Theorem 4.8) establishes a unique equilibrium in a neighbourhood of the frictionless equilibrium, provided the agents’ risk aversions are sufficiently similar (|gamma^1 - gamma^2| is small enough). The proof is based on a novel Picard iteration scheme: for a given volatility process, the FBSDE for optimal positions and trading rates can be solved via stochastic Riccati equations (Lemma 3.2, equation 3.5), and stability estimates (Section 7) show that the mapping from volatility to volatility (via equilibrium) is a contraction.

The benchmark example with linear state dynamics (Section 5) yields a system of four coupled Riccati ODEs (Proposition 5.1) that can be solved numerically. The small-cost asymptotics (Section 5.3) reveal that illiquidity discounts grow as sqrt(lambda)*T, the liquidity premium in expected returns scales as (gamma^1 - gamma^2) * sigma^2 * a^2 * (phi^1 - phi_bar^1), and the volatility correction B(t) has the same sign as gamma^1 - gamma^2 — so transaction costs increase volatility when the more risk-averse agent is the “trend follower.”

Key Contributions

  • First equilibrium existence result with endogenous volatility and quadratic transaction costs
  • Novel well-posedness theory for fully coupled nonlinear FBSDEs (Theorem 4.8, Theorem 8.3)
  • Picard iteration via stochastic Riccati equations: one-dimensional iteration on the volatility process only
  • Stability estimates in BMO spaces (Section 7, Lemmas 7.1-7.5) for the BSRDE, target process, and optimal strategy
  • Illiquidity discount, liquidity premium, and volatility correction all have the same sign, determined by (gamma^2 - gamma^1)
  • Positive relationship between transaction costs and volatility, corroborating empirical evidence

Methodology

Individual optimality reduces to a quadratic tracking problem (equation 3.2) solved by the BSRDE (3.5) for the tracking speed c_t. The optimal strategy tracks the frictionless target at a (random) speed determined by c_t. Market clearing couples agents’ FBSDEs into the system (4.6-4.8). The key innovation is the Picard iteration on volatility: given sigma, solve for the BSRDE c, the target xi-bar, and the optimal strategy phi; then compute the implied volatility from the BSDE (4.8); iterate. Contraction in the H^2_BMO norm yields convergence for sufficiently similar risk aversions.

Key Findings

  • Transaction costs create endogenous mean-reversion in expected returns AND endogenous volatility changes
  • Illiquidity discount scales as sqrt(lambda) * T for large T (equation 5.6)
  • Liquidity premium and volatility correction always have the same sign as (gamma^2 - gamma^1)
  • If gamma^2 > gamma^1 (“trend follower” is more risk-averse), transaction costs increase volatility — consistent with empirical evidence
  • The equilibrium is unique in a ball B_infinity(R) around the frictionless equilibrium, with explicit bound on R

Important References

  1. Equilibrium Returns with Transaction Costs — Herdegen, Muhle-Karbe (2018), the exogenous-volatility precursor
  2. Asset Pricing with General Transaction Costs — Gonon, Muhle-Karbe, Shi (2020), general cost functions
  3. Investing with Liquid and Illiquid Assets — Garleanu and Pedersen (2013), single-agent quadratic costs

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